作用在圆上的群的不变拟同构与SCL的非等价性
Invariant quasimorphisms for groups acting on the circle and non-equivalence of SCL
摘要：
我们构造了作用在圆上的群的不变拟同构。此外，我们提供了一个标准的不可扩展的拟同构和一个明确的公式，我们的拟同构的值的庞加莱平移数。利用它们，我们证明了对于亏格至少为$2$的面群$G=π_1(Σ_{\ell})$及其换位子群$N = [π_1(Σ_{\ell}), π_1(Σ_{\ell})]$，稳定换位子长$\mathrm{scl}_G$与稳定混合换位子长$\mathrm{scl}_{G,N}$不是双Lipschitz等价的.我们还证明了一对$(G,N)$的不等价性，使得$G$是一个$3$维闭双曲映射环面的基本群。这些对用作这种$(G,N)$的第一族示例，其中$G$是并行生成的。
英文摘要：
We construct invariant quasimorphisms for groups acting on the circle. Furthermore, we provide a criterion for the non-extendablity of the resulting quasimorphisms and an explicit formula which relates the values of our quasimorphisms to those of the Poincaré translation number. By using them, we show that the stable commutator length $\mathrm{scl}_G$ and the stable mixed commutator length $\mathrm{scl}_{G,N}$ are not bi-Lipschitzly equivalent for the surface group $G=π_1(Σ_{\ell})$ of genus at least $2$ and its commutator subgroup $N = [π_1(Σ_{\ell}), π_1(Σ_{\ell})]$. We also show the non-equivalence for a pair $(G,N)$ such that $G$ is the fundamental group of a $3$-dimensional closed hyperbolic mapping torus. These pairs serve as the first family of examples of such $(G,N)$ in which $G$ is finitely generated.
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